Congruence ABC implies ABC

1999/09/17 by Jordan S. Ellenberg, Ellenberg, Jordan S.
Mathematics · #11D41 (Primary) 11D75 (Secondary) #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11D41 #msc:11D75

paper · pdf · doi:10.48550/arxiv.math/9909098

4 pages; to appear, Indag. Math

arxiv created 1999/09/17 · arxiv updated 2009/11/30

Abstract

The ABC conjecture of Masser and Oesterle' states that if (a,b,c) are coprime integers with a + b + c = 0, then sup(|a|,|b|,|c|) < c_e (rad(abc))^1+e for any e > 0. Oesterle' has observed that if the ABC conjecture holds for all (a,b,c) with 16 | abc, then the full ABC conjecture holds. We extend that result to show that, for every integer N, the "congruence ABC conjecture" that ABC holds for all (a,b,c) with N|abc implies the full ABC conjecture.

Related

• Applications of the abc conjecture to powerful numbers
• Squarefree integers and the abc conjecture
• The abc conjecture is true almost always
• A more general abc conjecture
• Baker's Explicit abc-Conjecture and applications